Application of the LCM to gear networks

Imagine we've interlocked two gears, one smaller, with 16 teeth, and one larger, with 32 teeth. We draw an arrow from the center of each gear to its point of contact with the other gear, so that the two arrow tips meet exactly. How many full rotations would we have to turn the smaller gear for the arrows to meet a second time? How many full rotations would we have to turn the larger gear? What is the answer to both of the previous questions if the smaller gear had 24 teeth instead? Generalizing this, if we had two gears, one smaller gear with \(g_1\) teeth, and another larger gear with \(g_2\) teeth, what formula tells us the number of full rotations necessary for the arrows to meet a second time? What if we didn't know which gear was larger a priori? What would our formula be then? That is, we're always turning \(g_1,\) but \(g_1\) could be smaller than \(g_2\) or vice-versa.

Generalizing even further, what if we had three gears, and two pairs of arrows? The first pair of arrows are black. They're between the first and second gear. The second pair of arrows are red. They're between the second and third gear. If we turn a gear at either end, how many rotations are necessary for both pairs of arrows to meet a second time? What if we turned the middle gear instead?

What if our chain consisted of not 3 gears, but \(n,\) and we've decided to turn gear \(i?\)

Next, make several circular networks of gears. For each network, rotate some single gear. After doing this for several networks, what do you notice? Now consider your previous formula for a chain of \(n\) gears, rotating gear \(i.\) Does the same formula work? And if not, what's the formula for the case of circular networks?

What about an arbitrary network of gears, where arrows are drawn between each pair of gears?